ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcom2 Structured version   GIF version

Theorem sbcom2 1860
Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof modified to be intuitionistic by Jim Kingdon, 19-Feb-2018.)
Assertion
Ref Expression
sbcom2 ([w / z][y / x]φ ↔ [y / x][w / z]φ)
Distinct variable groups:   x,z   x,w   y,z
Allowed substitution hints:   φ(x,y,z,w)

Proof of Theorem sbcom2
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 sbcom2v2 1859 . . . 4 ([v / z][y / x]φ ↔ [y / x][v / z]φ)
21sbbii 1645 . . 3 ([w / v][v / z][y / x]φ ↔ [w / v][y / x][v / z]φ)
3 sbcom2v2 1859 . . 3 ([w / v][y / x][v / z]φ ↔ [y / x][w / v][v / z]φ)
42, 3bitri 173 . 2 ([w / v][v / z][y / x]φ ↔ [y / x][w / v][v / z]φ)
5 ax-17 1416 . . 3 ([y / x]φv[y / x]φ)
65sbco2v 1818 . 2 ([w / v][v / z][y / x]φ ↔ [w / z][y / x]φ)
7 ax-17 1416 . . . 4 (φvφ)
87sbco2v 1818 . . 3 ([w / v][v / z]φ ↔ [w / z]φ)
98sbbii 1645 . 2 ([y / x][w / v][v / z]φ ↔ [y / x][w / z]φ)
104, 6, 93bitr3i 199 1 ([w / z][y / x]φ ↔ [y / x][w / z]φ)
Colors of variables: wff set class
Syntax hints:  wb 98  [wsb 1642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  2sb5rf  1862  2sb6rf  1863  sbco4lem  1879  sbco4  1880  sbmo  1956  cnvopab  4669
  Copyright terms: Public domain W3C validator